Audition-style fade handles in Premiere

Agree, FCPX makes this super easy. I keep adding default audio transitions and it always adds as a crossfade to the clip, so whenever I move the clip it breaks the crossfade. FCPX has the handle that just pulls out from the clip and follows the clip wherever it goes. You can keyframe and everything separately from it. It saves me so much time on the pops in every clip that happens.

This would be a great addition. Its much easier than current fade transition, as sometimes we work with maximized timeline view. Would be nice if added to the video clips too.

and some automatic fades/crossfades would be also great

This would be great. One thing you should try out is binding "Apply Default Audio Transition To Playhead" and "Apply Default Audio Transition From Playhead"
This is the fastest way I've found to set audio fade in and outs. It works both on selected clips or selected tracks. There is a similar keybind for video as well.

Fcpx has this and it is very fast to use.